IDENTIFICATION OF INCOME-LEISURE PREFERENCES
AND EVALUATION OF INCOME TAX POLICY
Charles F. Manski
Department of Economics and Institute for Policy Research
Northwestern University
first public version: January 2012
this version: March 2012
Abstract
The merits of alternative income tax policies depend on the population distribution of preferences for income
and leisure. Standard theory, which supposes that persons want more income and more leisure, does not
predict how they resolve the tension between these desires. Empirical studies of labor supply have imposed
strong preference assumptions that lack foundation. This paper examines anew the problem of inference on
income-leisure preferences and considers the implications for evaluation of tax policy. I first perform a basic
revealed-preference analysis assuming only that persons prefer more income and leisure. This shows that
observation of a person’s time allocation under a status quo tax policy may bound his allocation under a
proposed policy or may have no implications, depending on the tax schedules and the person’s status quo
time allocation. I next explore the identifying power of two classes of assumptions that restrict the
distribution of income-leisure preferences. One assumes that groups of persons who face different choice
sets have the same preference distribution. The second restricts the shape of this distribution. The generic
finding is partial identification of preferences. This implies partial prediction of tax revenue under proposed
policies and partial knowledge of the welfare function for utilitarian policy evaluation.
This research was supported in part by National Science Foundation grant SES-0911181. I am grateful to
Matthew Masten for excellent research assistance and to Jörg Stoye for his sharing of computer code used
in the computational experiments of Section 3. I am grateful to Masten, Stoye, and Richard Blundell for
helpful comments and discussions. I have benefitted from the opportunity to present this work at the
December 2011 UCL Workshop on Consumer Behaviour and Welfare Measurement and in seminars at
Northwestern, Pittsburgh, and Yale.
1. Introduction
Economists have long recognized that the relative merits of alternative income tax policies depend
on the preferences of individuals for income and leisure. The profession has also recognized the difficulty
of inference on these preferences. In the conclusion to his seminal study of optimal income taxation, James
Mirrlees wrote (Mirrlees, 1971, p. 207): “The examples discussed confirm, as one would expect, that the
shape of the optimum earned-income tax schedule is rather sensitive to the distribution of skills within the
population, and to the income-leisure preferences postulated. Neither is easy to estimate for real economies.”
Income-leisure preferences play both positive and normative roles in analysis of tax policy, and it
is important to distinguish them. The positive role is that preferences yield labor supply and other decisions
that determine tax revenue. The normative role is that social welfare aggregates individual preferences in
utilitarian policy evaluation. Thus, comparison of tax policies requires knowledge of preferences both to
predict tax revenues and to compute the welfare achieved by alternative policies.
As I see it, we lack the knowledge of preferences necessary to credibly evaluate income tax policies.
Analysis of the identification problem in Sections 2 and 3 will explain how I have reached this sober
conclusion. As prelude, I first offer my perspective on research studying the response of labor supply to
taxation, a subject that has receive enormous attention. I then outline the paper.
Taxation and Labor Supply
To begin, recall that standard economic theory does not predict the response of labor supply to
income taxation. To the contrary, it shows that a worker may rationally respond in disparate ways. As tax
rates increase, a person may rationally decide to work less, work more, or not change his labor supply at all.
The silence of theory on labor supply has long been appreciated; see Robbins (1930). Modern labor
economics envisions labor supply as a complex sequence of schooling, occupation, and work effort decisions
made under uncertainty over the life course. However, we need only consider a simple static scenario to see
2
that a person may respond to income taxes in disparate ways.
Suppose that a person with a predetermined wage and no unearned income allocates each day
between paid work and the various non-paid activities that economists have traditionally called leisure. Let
a proportional income tax reduce his wage by the prevailing tax rate, yielding his net wage. Assume that the
person allocates time to maximize utility, which is an increasing function of net income and leisure.
Different utility functions imply different relationships between the tax rate and labor supply. The
labor supply implied by utility functions in the Constant-Elasticity-of-Substitution (CES) family increases
or decreases with the tax rate depending on the elasticity of substitution. Other utility functions imply that
labor supply is backward-bending. That is, hours worked may initially increase as net wage rises from zero
but, above some threshold, decrease as net wage rises further. Still other utility functions yield more
complex non-monotone relationships between net wage and labor supply. The review article of Stern (1986)
describes a broad spectrum of possibilities.
Given that theory does not predict how income taxation affects labor supply, prediction requires
empirical analysis. Robbins (1930) emphasized this, concluding his article as follows (p. 129): “we are left
with the conclusion . . . . that any attempt to predict the effect of a change in the terms on which income is
earned must proceed by inductive investigation of elasticities.”
Economists have performed numerous empirical studies of labor supply. Their methodologies, data,
and findings have been summarized and critiqued in multiple lengthy review articles including Pencavel
(1986), Killingsworth and Heckman (1986), Blundell and MaCurdy (1999), Meghir and Phillips (2010),
Keane (2011), and Saez, Slemrod, and Giertz (2011). Attempting to distill the huge literature, Meghir and
Phillips (2010) write (p. 204): “Our conclusion is that hours of work do not respond particularly strongly to
the financial incentives created by tax changes for men, but they are a little more responsive for married
women and lone mothers. On the other hand, the decision whether or not to take paid work at all is quite
sensitive to taxation and benefits for women and mothers in particular.” Saez, Slemrod, and Giertz (2011)
3
similarly write (p. 1): “With some exceptions, the profession has settled on a value for this elasticity close
to zero for prime-age males, although for married women the responsiveness of labor force participation
appears to be significant. Overall, though, the compensated elasticity of labor appears to be fairly small.”
Keane (2011) expresses a different perspective, writing (p. 1071): “My review suggests that labor supply of
men may be more elastic than conventional wisdom suggests.”
Reading the recent empirical literature, I have been struck to find that while authors may differ on
the magnitude of labor-supply elasticities, they largely agree on the sign. The consensus is that increasing
tax rates usually reduces work effort. Considering the effect of a rise in a proportional tax, Meghir and
Phillips (2010) write (p. 207): “in most cases this will lead to less work, but when the income effect
dominates the substitution effect at high hours of work it may increase effort.” Keane (2011) states the
directionality of the effect without reservation, writing (p. 963): “the use of labor income taxation to raise
revenue causes people to work less.” Here and elsewhere, empirical researchers may recognize the
theoretical possibility that effort may increase with tax rates but view this as an empirical rarity rather than
a regularity. This view has been accepted in official government forecasts of the response of labor supply
to income taxation. See Congressional Budget Office (2007).1
Examining the models of labor supply used in empirical research, I have become concerned that the
prevailing consensus on the sign of elasticities may be an artifact of model specification rather than an
expression of reality.2 Models differ across studies, but they generally share two key assumptions. First,
they suppose that labor supply varies monotonically with net wages. Thus, model specifications do not
1
Curiously, the opposite consensus prevailed early in the twentieth century. Gilbert and Pfouts
(1958) cite assertions by Pigou and Knight in the 1920s that increasing tax rates increases work effort.
2
The literature contains some precedent for my concern that empirical findings on labor supply may
be artifacts of model specification. Concluding his detailed comparison of alternative labor supply functions,
Stern (1986) wrote (p. 173): “Our general conclusion must be in favour of diversity of functions and great
caution in drawing policy conclusions on results based on a particular form.” Stern and other writers such
as Blundell and MaCurdy (1999) have called attention to the potential detrimental consequences of restrictive
functional-form assumptions.
4
generally permit backward-bending labor supply functions or other non-monotone relationships. Second,
they suppose that the response of labor supply to net wage is homogeneous within broad demographic
groups. With occasional exceptions, researchers specify hours-of-work equations that permit hours to vary
additively across group members but that assume constant treatment response. That is, they assume that all
group members would adjust hours worked in the same way in response to a conjectural change in net wage.3
The reality may be that persons have heterogeneous income-leisure preferences and, consequently,
heterogeneous labor-supply functions. Some may increase work effort with net wage, others may decrease
effort, and still others may exhibit a non-monotone wage-effort relationship. If so, estimates of models that
assume monotonicity and homogeneity of labor supply can at most characterize the behavior of an artificial
“representative” person. The estimates may not have even this limited interpretation.
3
The review article of Keane (2011) uses this hours-of-work equation to illustrate common practices
in the literature (p. 972):
ln hit = â + e@ln[wit(1 ! ôt)] + âI@Nit + åit.
Here i denotes a person, t is a time period, h is hours worked, w is wage, ô is a proportional tax rate, N is
unearned income, and (â, e, âI) are parameters. Keane writes: “Crucial is the addition of the stochastic term
åit, which enables the model to explain heterogeneity in behavior.” However, the model permits only a very
restricted form of heterogeneity. Hours of work may vary additively across persons through åit but the
parameters (â, e, âI) do not vary across persons. In particular, the elasticity parameter e expresses the
assumption of constant treatment response.
The occasional exceptions to such strong homogeneity assumptions begin with the important early
research of Burtless and Hausman (1978), who emphasized the potential importance of heterogeneity but
were more limited in their ability to operationalize it. The authors ultimately assumed homogeneous
response of hours worked to changes in net wage. The recent contributions of Blundell and Shephard (2012)
and Blundell, Bozio, and Laroque (2011) also emphasize heterogeneity, and are able to do more to
operationalize it within a parametric random utility framework.
An exception to parametric modeling is Blomquist and Newey (2002). They initially propose general
nonparametric mean regression of hours worked on wage and on tax schedule variables that characterize a
person’s (income, leisure) choice set. Concerned with the dimensionality of the variables needed to
characterize the choice set, they subsequently study estimation of a model that takes desired hours of work
to be a nonparametric but increasing function of a scalar measure of preference heterogeneity.
5
This Paper
In light of the above, this paper examines anew the problem of identification of income-leisure
preferences and draws implications for evaluation of tax policy. I study inference when data on time
allocation under a status quo tax policy are interpreted through the lens of standard theory. To illuminate
elemental issues, I find it productive to study the classical static model in which persons must allocate one
unit of time to work and leisure in an environment with predetermined production of public goods.
The classical static model is obviously simplistic, but it serves well the purpose of showing the
difficulty of inference on preferences. Inference becomes even more difficult when one considers models
of dynamic labor supply (e.g., MaCurdy 1985), models that consider how income taxation may affect
decisions other than time allocation (e.g., Feldstein, 1995), models that conjecture socially interdependent
preferences (e.g., Lindbeck, Nyberg, and Weibull, 1999), or models that suppose persons may be boundedly
rational (e.g., Chetty, 2009). Inference also becomes more difficult when one wants to study policies that
vary production of public goods.4 These developments all generalize the classical static model. Hence, they
logically pose more difficult inferential problems.
Sections 2 and 3 consider the use of revealed preference analysis to predict labor supply and tax
revenue under a proposed policy that would alter the status quo tax schedule. A central objective is to shed
light on how maintained assumptions affect the conclusions that one may draw. As in my past research, I
find it illuminating to begin with weak assumptions and then to characterize the identifying power of stronger
assumptions. I have exposited this approach to empirical research in Manski (2007a) and elsewhere.
Section 2 assumes only that persons prefer to have more income and leisure. Basic revealed-
4
The literature on income taxation largely abstracts from this important question. Over thirty years
ago, Wildasin (1979) observed (p.63-64): “The proper way of taking the effects of distortionary taxes into
account in evaluating public expenditure depends sensitively on complement-substitute relations between
public and private goods.” He went on (p. 64): “Most bothersome of all is the fact that we have very little
empirical information on the interaction between public good provision and private demand.” This
observation remains accurate today.
6
preference analysis then shows that observation of a person’s time allocation under a status quo tax policy
may bound his allocation under a proposed policy or may have no implications, depending on the tax
schedules and the person’s status quo time allocation. Basic analysis assuming only that more-is-better
generically does not predict the sign of labor-supply response to change in the tax schedule.
Section 3 explores the identifying power of assumptions restricting the population distribution of
preferences. For this analysis, I find it analytically helpful to suppose that persons choose among a finite set
of feasible (income, leisure) values rather than the continuum often assumed in the literature. I then use the
discrete-choice framework of Manski (2007b) to characterize preferences and to predict aggregate labor
supply and tax revenue when various assumptions restrict the distribution of preferences. I show how to
determine the identifying power of two classes of assumptions. The first assumes that groups of persons who
face different choice sets and/or have different covariates have the same distribution of preferences. The
second restricts the shape of this distribution. The generic finding is partial identification of the preference
distribution.
This implies partial ability to predict tax revenue under proposed policies.
I use a
computational experiment to illustrate.
The analysis of Sections 2 and 3 reaches highly cautionary conclusions about present knowledge of
income-leisure. The concluding Section 4 discusses some implications for evaluation of tax policy. A
familiar exercise in normative public economics poses a utilitarian social welfare function and ranks tax
policies by the welfare they achieve. This requires knowledge of income-leisure preferences to predict tax
revenues and compute welfare. I observe that partial knowledge of preferences implies two difficulties for
policy evaluation. One can only partially predict tax revenue and one can only partially evaluate the
utilitarian welfare of policies. We may potentially improve our capacity to evaluate tax policy by collecting
richer data that sheds more light on preferences. I pose some possibilities for data collection.
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2. Basic Analysis of Revealed Preference
To begin, Section 2.1 formalizes the conventional description of income taxation and labor supply
that I use. Section 2.2 applies basic revealed-preference analysis to derive partial predictions of individual
labor supply under counterfactual alternatives to a status quo tax policy. Section 2.3 studies simple cases
where the net-income functions under the status quo and proposed tax schedules cross once. I call this
section a “basic” analysis of revealed preference because I maintain no assumptions about preferences except
that individual utility is an increasing function of net income and leisure. In short, more is better.
In the absence of assumptions that restrict the population distribution of preferences, predicting
population labor supply under a proposed tax policy simply requires aggregation of individual predictions.
Hence, the analysis below focuses on one person.
2.1. Tax Policy and Labor Supply
Suppose that a person labeled j is endowed with wage wj, unearned income zj, and one unit of time.
The person must allocate his time endowment between leisure and work. If he allocates a fraction L 0 [0,
1] to leisure and 1 ! L to work, he receives gross income wj(1 ! L) + zj.
The status quo tax policy, denoted S, subtracts the work-dependent tax revenue RjS(L) from gross
income, leaving j with net income
(1)
YjS(L) / wj(1 ! L) + zj ! RjS(L).
The RjS(@) notation allows the tax schedule to be specific to person j. Thus, the tax may depend on unearned
income and on personal attributes that determine eligibility for exemptions and deductions. Taxes may take
8
positive or negative values.
Person j chooses a value of L from a set Ëj d [0, 1] of feasible leisure alternatives. Most analysis
of labor supply supposes that Ëj = [0, 1]. However, it may be more realistic to suppose that only a few
allocations are feasible. For example, Ëj = {0, ½, 1} means that the feasible options are full-time work (L
= 0), half-time work (L = ½), and no work (L = 1).
Preferences are expressed in person j’s utility function Uj(@, @), whose arguments are (net income,
leisure). Utility is strictly increasing in both arguments. Let LjS 0 Ëj denote the amount of leisure that j
chooses under tax schedule RjS(@). Utility maximization implies these inequalities:
(2)
Uj[YjS(LjS), LjS] $ Uj[YjS(L), L], all L 0 Ëj.
Using Uj(@, @) to express preferences suppresses the possible dependence of preferences on the public
goods produced with tax revenue under policy S. This is innocuous if the objective is to predict labor supply
under an alternative policy that alters the tax schedule but leaves public-good production unchanged.
2.2. Predicting Labor Supply under a Proposed Tax Schedule
Suppose that one observes the wage, unearned income, and other tax-relevant attributes of person
j. One also observes the leisure LjS chosen by j under tax schedule RjS(@). Let RjT(@) denote the tax schedule
if j were to face a proposed tax policy T. What can one predict about time allocation under RjT(@)? The
answer depends on the value of LjS and on the budget sets {[YjS(L), L], L 0 Ëj} and {[YjT(L), L], L 0 Ëj} that
j faces under the status-quo and proposed tax schedules.
9
Illustration
I begin with an illustration, which modestly extends the original revealed-preference argument of
Samuelson (1938). Consider the tax schedules in Figure 1. Policy S has a two-rate progressive schedule and
T has a proportional one, the latter crossing the former from above when leisure equals L*j . Person j has no
unearned income.
Figure 1: Net Income with Progressive and Proportional Tax Schedules
Suppose that LjS 0 [0, L*j ], and consider any feasible L > L*j . The (income, leisure) pair [YjT(LjS), LjS]
is feasible under policy T. Since more income is better than less, this pair is preferred to the pair [YjS(LjS),
LjS] chosen under S. Pair [YjS(L), L] is feasible under policy S. Since more income is preferred to less, this
pair is preferred to [YjT(L), L]. The observation that j chose [YjS(LjS), LjS] reveals that j prefers this pair to
[YjS(L), L]. Combining these preference inequalities implies that he prefers [YjT(LjS), LjS] to [YjT(L), L].
Thus, if person j were to face the proposed tax schedule, he would not choose any L > L*j . It will be shown
10
below that this exhausts the predictive power of basic revealed preference analysis.5
General Analysis
I now formally perform basic revealed-preference analysis without restricting the tax schedules
generated by the status quo and proposed policies. The analysis requires a few definitions.
Given two bivariate real vectors (a, b) and (aN, bN), define (a, b) < (aN, bN) and say that (a, b) is strictly
smaller than (aN, bN) if (a # aN and b # bN) and (a < aN or b < bN). Define (a, b) # (aN, bN) and say that (a, b)
is weakly smaller than (aN, bN) if (a # aN and b # bN). Define strictly and weakly larger analogously.
Let Ëj< denote the feasible leisure values such that the (income, leisure) pairs in the RjT(@) budget set
are strictly smaller than some pair in the RjS(@) budget set. That is,
(3)
Ëj< / {L< 0 Ëj: [YjT(L< ), L< ] < [YjS(L), L] for some L 0 Ëj}.
Let Ëj> denote the leisure values such that the (income, leisure) pairs in the RjT(@) budget set are strictly larger
than [YjS(LjS), LjS], the pair that j chooses under the status quo policy. That is,
(4)
Ëj> / {L> 0 Ëj: [YjT(L> ), L> ] > [YjS(LjS), LjS]}.
Define Ëj# and Ëj$ analogously, with weak inequalities replacing the strict ones. The sets (Ëj<, Ëj>, Ëj#, Ëj$)
are functions of the tax schedules RjS(@) and RjT(@), but I keep this implicit to simplify the notation.
Revealed preference analysis combines three facts. First, choice of LjS yields the inequalities (2).
5
I have written that this derivation modestly extends Samuelson (1938). Samuelson considered use
of data on the commodity bundle chosen under a status quo linear budget set to predict the bundle that would
be chosen under a counterfactual linear budget set. Figure 1 differs only in that the budget set produced by
policy T is not linear. Linearity of either budget set is immaterial. Samuelson’s argument holds whenever
the status quo and counterfactual budget sets cross one another.
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Second, the assumption that more is better implies that, for each L< 0 Ëj< , there exists an L 0 Ëj such that
(5)
Uj[YjS(L), L] > Uj[YjT (L< ), L< ].
Similarly, for each L# 0 Ëj# , there exists an L 0 Ëj such that
(5N)
Uj[YjS(L), L] $ Uj[YjT(L# ), L# ].
Third, more-is-better implies that for each L> 0 Ëj> ,
(6)
Uj[YjT(L> ), L> ] > Uj[YjS(LjS), LjS].
Similarly, for each L$ 0 Ëj$ ,
(6N)
Uj[YjT(L$ ), L$ ] $ Uj[YjS(LjS), LjS].
Combining (2), (5), and (6N) yields
(7)
Uj[YjT (L$ ), L$ ] $ Uj[YjS(LjS), LjS] > Uj[YjT(L< ), L< ],
all L< 0 Ëj< and L$ 0 Ëj$ .
Similarly, combining (2), (5N), and (6) yields
(7N)
Uj[YjT(L> ), L> ] > Uj[YjS(LjS), LjS] $ Uj[YjT(L#), L#],
all L# 0 Ëj# and L> 0 Ëj>.
12
Proposition 1 summarizes the implications for prediction of time allocation.
Proposition 1: Let Ëj< and Ëj$ be non-empty. Then if person j were to face tax schedule RjT(@), he would not
choose any leisure value in Ëj<. Let Ëj# and Ëj> be non-empty. Then if j were to face RjT(@), he would not
choose any leisure value in Ëj# .
~
These conclusions exhaust the predictive power of basic revealed preference analysis for time allocation
under the proposed tax schedule.
Welfare Implications
We may also draw partial conclusions about the welfare implications of replacing the status quo tax
schedule with the proposed one. If Ëj< = Ëj, replacement of the status quo tax schedule with the proposal
strictly decreases the welfare of person j. This follows purely from the assumption that more is better and
does not require observation of j’s time allocation under the status quo. If Ëj> is non-empty, replacement of
the status quo with the proposal strictly increases welfare. This holds because non-emptiness of Ëj> means
that the budget set under the proposed schedule contains an (income, leisure) pair larger than the one j
obtained under the status quo. Basic analysis yields no welfare conclusions if Ëj< is a proper subset of Ëj
and Ëj> is empty.
2.3. Prediction When Downward-Sloping Net-Income Functions Cross Once
The above analysis was entirely general. I henceforth suppose that the net income functions YjS(@)
and YjT (@) implied by the status quo and proposed tax schedules are downward sloping. This condition is
substantively innocuous, it being violated only if the marginal tax rate is above 100 percent at some level of
13
work effort.
Given that the schedules are downward sloping, basic revealed preference analysis can have
predictive power for labor supply only if the two net-income functions cross at least once. Formally, YjT(@)
crosses YjS(@) from above at some L*j 0 (0, 1) if YjT(L*j ) = YjS(L*j ), YjT(L) > YjS(L) for L < L*j , and YjT(L) <
YjS(L) for L > L*j . Similarly, YjT (@) crosses YjS(@) from below at L*j if the inequalities are reversed.
To see that a crossing is necessary for predictive power, consider a policy T such that YjT(L) $ YjS(L)
for all L 0 [0, 1]. Thus, at all L a person pays weakly less tax under T than S. Then set Ëj< is empty because
YjT(L< ) < YjS(L) Y L< > L. Similarly, consider T such that YjT(L) # YjS(L) for all L 0 [0, 1]. Thus, at all
L a person pays weakly more tax under T than S. Then Ëj$ is empty because YjT(L< ) > YjS(LjS) Y L< < LjS.
Policy comparison is simple when the two tax schedules cross exactly once, as in Figure 1. When,
as in the figure, YjT(@) crosses YjS(@) from above, Ëj< = [L 0 Ëj: L > L*j ] and Ëj# = [L 0 Ëj: L $ L*j ].6 The set
Ëj> is non-empty if and only if LjS < L*j , and Ëj$ is non-empty if and only if LjS # L*j .7 Hence, observation that
LjS < L*j implies that if person j were to face tax schedule RjT(@), he would not choose any L $ L*j .
Observation that LjS # L*j implies that he would not choose any L > L*j . Observation that LjS > L*j has no
predictive power.8
To see this, first observe that L > L*j Y [YjT(L), L] < [YjS(L), L] and L = L*j Y [YjT(L), L] = [YjS(L),
L]. Hence, [L 0 Ëj: L > L*j ] d Ëj< and [L 0 Ëj: L $L*j ] d Ëj# . Now consider L < L*j . Then YjT(L) > YjS(L).
Given that YjS(@) is downward sloping, YjT (L) < YjS (LN) Y LN < L. Hence, there exists no LN such that
[YjT(L), L] < [YjS(LN), LN]. Hence, L < L*j Y L ó Ëj<.
6
To see this, first observe that LjS < L*j Y [YjT(LjS), LjS] > [YjS(LjS), LjS] and LjS = L*j Y [YjT(LjS), LjS]
= [YjS(LjS), LjS]. Hence, LjS < L*j implies that Ëj> is non-empty and LjS = L*j implies that Ëj$ is non-empty.
Now consider LjS > L*j . Then YjT(LjS) < YjS(LjS). Given that YjT(@) is downward sloping, YjT(LN) $ YjS(LjS)
Y LN < LjS. Hence, there exists no LN such that [YjT(LN), LN] $ [YjS(LjS), LjS]. Hence, LjS > L*j implies that
Ëj$ is empty.
7
Analogous findings hold when YjT(@) crosses YjS(@) from below. Then Ëj< = [L< 0 Ëj: L < L*j ] and
Ëj# = [L# 0 Ëj: L # L*j ]. The set Ëj> is non-empty if LjS > L*j and Ëj$ is non-empty if LjS $ L*j . Hence,
observation that LjS > L*j implies that if j were to face schedule RjT(@), he would not choose any L # L*j .
Observation that LjS $ L*j implies that he would not choose any L < L*j . Observation that LjS < L*j has no
predictive power.
8
14
The above shows that, when the proposed tax schedule crosses the status quo from above, the sign
of the response of labor supply to the change in tax schedule is predictable only in two cases. If a person
works full time under the status quo schedule, then obviously labor supply cannot increase under the
proposed schedule. If status quo leisure is LjS = L*j , then basic analysis of revealed preference predicts that
labor supply will not decrease under the proposed schedule. Thus, basic analysis predicts the sign of the
labor-supply response only in the latter very special case.
Numerical Illustration
To illustrate, suppose that the progressive schedule taxes income at rate 15 percent up to $50,000
per year and at rate 25 percent above $50,000. Let the proportional schedule tax all income at rate 20
percent. Thus, the tax schedules cross when gross income equals $100,000, when both take $20,000 tax and
yield net income $80,000. As in Figure 1, suppose that person j has no unearned income. The conclusions
that may be drawn about time allocation and welfare under the proposed tax schedule depend on person j’s
wage, his feasible set of time allocations, and the allocation he chooses under the status quo schedule. In
what follows, I measure wage as the annual gross labor income that j would receive for full-time work and
I suppose that all time allocations are feasible.
Suppose first that w < $100,000. Then net income under the proposed schedule is lower than net
income under the status quo schedule for all L 0 [0, 1]. Hence, the proposed schedule definitely lowers
welfare. Revealed preference analysis yields no prediction of time allocation.
Next let w = $150,000. Then L* = 1/3, which is the roughly the case in Figure 1. Basic revealed
preference analysis has predictive power if LjS # 1/3; that is, if j works at least 2/3 of the time in the status
quo. Then we can conclude that leisure under the proposed schedule would lie in the interval [0, 1/3] and
that welfare would increase. However, no prediction of time allocation or welfare is possible if LjS > 1/3.
Finally let w = $500,000. Then L* = 4/5. If LjS # 4/5, we can conclude that leisure under the
15
proposed schedule would lie in the interval [0, 4/5] and that welfare would increase. No prediction is
possible if LjS > 4/5.
Inequalities on Aggregate Labor Supply
Although basic analysis rarely predicts the sign of individual labor-supply response, it generically
yields some simple inequalities relating population labor supply under the status quo and proposed tax
policies. Let the proposed tax schedule cross the status quo from above. Then, among persons whose tax
schedules cross at the same point R, the fraction who choose leisure less than R is higher under the proposed
schedule than under the status quo schedule.
To see this, let J denote a population of persons formalized as a probability space (J, Ù, P). Let LjT
denote the leisure that person j would choose under the proposed schedule RjT(@). We have found that, for
each j 0 J, LjS < L*j Y LjT < L*j and LjS # L*j Y LjT # L*j . It follows that, for all R 0 [0, 1],9
(8a)
P(LS < L**L* = R) # P(LT < L**L* = R)
(8b)
P(LS # L**L* = R) # P(LT # L**L* = R).
Inequalities (8a)-(8b) formalize a rather limited sense in which decreasing the progressivity of a tax
schedule must weakly increase aggregate labor supply. If schedule T crosses S from above at point R, then
T is less progressive than S in the specific sense that, among people with crossing point R, T yields higher
net income than S when leisure is less than R and lower net income than S when leisure is larger than R.
I say that the inequalities formalize a “rather limited” relationship between progressivity and labor
9
Hoderlein and Stoye (2012), Lemma 1 reports a version of this inequality in the context of the
classical problem of consumer demand for two goods and cites earlier, more abstract, research that implies
the result. In the classical demand setting, one observes demand under status quo product prices and wants
to predict demand under counterfactual prices. Among persons with the same income, the status quo and
counterfactual budget lines necessarily cross once, at the same point.
16
supply because they hold only at the crossing point of the two tax schedules. Consider (8a). It is consistent
with rational choice and the more-is-better assumption to have the reverse inequality hold at any L other than
R. That is, it can happen that P(LS < L*L* = R) > P(LT < L*L* = R) for any L … R.
3. Restrictions on the Preference Distribution
A huge distance separates basic revealed preference analysis from the practice of empirical analysis
of labor supply. As noted in the Introduction, the models used in empirical studies usually suppose that labor
supply responds monotonically to variation in net wage. Moreover, it is common to assume that time
allocation differs across persons only via a person-specific additive constant.
To explore the vast middle ground between basic analysis and current practice, I find it productive
to use the discrete-choice framework developed in Manski (2007b). Section 3.1 lays out basic concepts and
notation. Section 3.2 shows how to determine the identifying power of alternative assumptions placing
restrictions on the population distribution of preferences. Section 3.3 discusses the mathematics and the
substance of the identification problem. Section 3.4 presents a computational experiment using the
framework to predict tax revenue under a proposed tax policy.
3.1. Preferences on a Finite Universe of Alternatives
Consider again a population J of persons formalized as a probability space (J, Ù, P). Suppose that
the status quo and proposed tax policies together generate a finite universe of potential values for (net
income, leisure). That is, the universe
17
(9)
A / {[YjS(L), L], [YjT(L), L], L 0 Ëj, j 0 J}
of distinct (Y, L) pairs that members of the population can realize under either policy has finite cardinality.
Current practice usually treats net income and leisure as continuous variables, but some authors have viewed
time-allocation as a choice among a discrete set of leisure options. See, for example, Blundell and Shephard
(2012), Hoynes (1996), and Keane and Moffitt (1998).
Viewing the universe of feasible (Y, L) pairs as finite rather than continuous enables one to explore
flexibly the identifying power of a spectrum of assumptions on the preferences of population members.
Analysis is most straightforward if almost all persons have strict preference orderings. There are *A*!
potential strict orderings on A. The number of feasible orderings is smaller than *A*! because the
assumption that more-is-better excludes some orderings, the number excluded depending on the composition
of A.10
Numerical Illustration
Consider again the illustration of Section 2.3. A status quo progressive schedule taxes income at rate
15 percent up to $50,000 per year and at rate 25 percent above $50,000. A proposed proportional schedule
taxes all income at rate 20 percent.
Consider a population of high-skilled, prime-age persons. Let each person have two time-allocation
alternatives, full-time or half-time work. Thus, Ëj = {0, ½} for all j 0 J. Let the population divide into two
groups with different annual wages for full-time work, w = $150,000 or w = $500,000. Then the (Y, L)
10
The assumption that almost all persons have strict preference orderings can be weakened. The
number of possible ordering is still finite, but larger, if we allow weak orderings that may exhibit indifference
among some (Y, L) pairs. However, when considering prediction, it is essential that at least some known
positive fraction of orderings be strict rather than weak. The concept of rationality makes no prediction
about choice among equally ranked alternatives. Revealed preference analysis is vacuous if one permits an
unknown fraction of the population to be indifferent among all elements of A.
18
choice sets are as follows:
Status Quo Schedule
Proposed Schedule
Group 1 (w = $150,000)
Full-time work
Half-time work
a
b
[117,500, 0]
[ 61,250, ½]
e
f
[120,000, 0]
[ 60,000, ½]
Group 2 (w = $500,000)
Full-time work
Half-time work
c
d
[380,000, 0]
[192,500, ½]
g
h
[400,000, 0]
[200,000, ½].
The universe A of potential values of (Y, L) contains eight elements, denoted a to h. Thus, the total
number of strict preference orderings is 8! = 40,320. However, the assumption that more-is-better eliminates
many orderings. In particular, it implies that feasible orderings satisfy the conditions g ™ c ™ e ™ a, h ™ d ™
b ™ f, and d ™ e. Eliminating the orderings that do not satisfy these conditions leaves 53 feasible orderings.
3.2. Identification Analysis
Let Ak, k 0 K denote the orderings of set A that are consistent with the assumption that more-isbetter. Then the population distribution of preferences is multinomial with at most *K* mass points, where
*K* # *A*!. All assumptions about population preferences take the form of restrictions on this multinomial
distribution.11
11
In Manski (2007b) I observed that prediction of counterfactual choice behavior does not require
that one distinguish all feasible preference orderings from one another. Multiple preference orderings may
yield identical choices when facing all status quo and proposed choice sets. Thus, it is only necessary to
distinguish groups of preference orderings that have different choice functions; that is, groups yielding
different choices in some relevant choice setting. This permits one to reduce the set K of distinct preference
orderings to a smaller set of distinct choice functions, which can be computationally advantageous in the
analysis below.
I will, however, not reduce K to the set of distinct choice functions. The reason is that prediction
of counterfactual choice behavior is not the sole concern of this paper. A further concern, discussed in
Section 4, is to compare policies via a utilitarian social welfare function. Such policy comparison requires
one to distinguish persons by their utility functions, not just by their choice functions.
19
I show here how to determine the identifying power of two layers of assumptions. The first assumes
that groups of persons who face different status quo or proposed choice sets have the same distribution of
preferences. The second adds restrictions on the shape of this distribution. In all that follows, I suppose that
one observes the wage, unearned income, and other tax-relevant attributes of the members of the population.
To begin, divide the population into a collection (Jm , m 0 M) of mutually exclusive and exhaustive
groups according to the (Y, L) choice sets they face under the status quo and proposed tax schedules. Thus,
all members of group m face a common choice set Cm d A of (Y, L) pairs under the status quo schedule and
a common set Dm d A under the proposed schedule. Groups m and mN are distinct if Cm … CmN or Dm … DmN.
Substantively, groups are composed of persons with the same tax-relevant attributes. Persons who have the
same wage, unearned income, exemptions/deductions, and live in the same tax jurisdiction face the same
income-leisure options.
Next, divide the population into a finite collection X of groups of persons with common observed
covariates. These covariates need not be tax relevant. For example, covariate groups may be defined by
gender, race, and/or education. Let Jmx denote the sub-population of persons who are members of tax group
m and covariate group x.
Let ðmx / (ðmxk, k 0 K) denote the multinomial distribution of preferences in (tax, covariate) group
(m, x). Thus, for each k 0 K, ðmxk is the fraction of group (m, x) who have preference ordering Ak. With no
assumptions on the population distribution of preferences, (ðmx, m 0 M, x 0 X) is an unrestricted set of
multinomial distributions, one for each (tax, covariate) group. The analysis of this section first assumes that
a specified set of groups have the same preference distribution and then places restrictions on the shape of
this distribution. Formally, consider
Assumption 1: For a specified class of (tax, covariate) groups N d M × X, there exists a single multinomial
distribution ðN / (ðNk, k 0 K) such that ðmx = ðN , all (m, x) 0 N.
~
20
Assumption 2: ðN lies in a specified set ÐN of multinomial distributions with at most *K* mass points.
~
Assumption 1 poses a substantive requirement whose credibility depends on the specification of N.
One may think it credible to assume that some (m, x) groups have the same preference distribution but not
others. Assumption 1 does not restrict the shape of the distribution ðN of (income, leisure) preferences.
Assumption 2 generically adds such restrictions. Particularly simple are linear restrictions that take ÐN to
be a set of multinomial distributions satisfying specified linear equalities or inequalities. These include
assumptions restricting the preference orderings that appear in the population. Suppose one believes that
some orderings, say KN0 d K, do not occur in class N. Then ðNk = 0, k 0 KN0 .
The identifying power of Assumptions 1 and 2 is easy to characterize, but we first need some new
notation. For each non-empty set B d A, let ck(B) denote the (Y, L) pair that a person with preference
ordering k would choose if he were to face choice set B. Let cj(B) denote the pair that person j would choose
and let the random variable c(B) denote the choice of a person randomly drawn from J. For each (m, x) 0
N and (Y, L) 0 Cm , let P[c(Cm ) = (Y, L)*Jmx ] give the observed fraction of sub-population Jmx who make
choice (Y, L) when facing their status quo choice set Cm . Similarly, for each (m, x) 0 N and (Y, L) 0 Dm ,
let P[c(Dm ) = (Y, L)*Jmx ] give the unobserved fraction of persons in Jmx who would choose (Y, L) if they were
to face choice set Dm .
Assumption 1 implies that the observed choice probabilities P[c(Cm ) = (Y, L)*Jmx], (Y, L) 0 Cm ,
(m, x) 0 N are related to the preference distribution ðN through the linear equations
(10) P[c(Cm ) = (Y, L)*Jmx ] =
3 1[ck(Cm ) = (Y, L)]@ðNk,
k0 K
(Y, L) 0 Cm , (m, x) 0 N.
For each value of (m, x) and (Y, L), the right-hand side adds up the fractions of Jmx whose preference
orderings make (Y, L) their most preferred alternative in Cm . Distribution ðN also satisfies the adding-up and
21
non-negativity conditions
(11a)
3 ðNk = 1,
k0K
(11b)
ðNk $ 0,
k 0 K.
Finally, Assumption 2 states that
(12)
ðN 0 ÐN .
Proposition 2 states the implications for identification of the distribution of preferences.
Proposition 2: Given Assumptions 1 and 2, the identification region for ðN is the set Ç(ðN ) of multinomial
distributions that solve (10), (11), and (12).
~
The set Ç(ðN ) has a simple structure when ÐN places linear restrictions on ðN . Then Ç(ðN ) is the
convex set of vectors in R*K* that solve a set of linear equalities and inequalities. Ç(ðN ) is non-empty if the
preference assumptions are correct; that is, if almost all persons have strict preferences satisfying more-isbetter and if Assumptions 1 and 2 hold. If Ç(ðN ) is empty, then some assumption is incorrect.
When Ç(ðN ) is non-empty, it is straightforward to predict the choice behavior of class N under the
proposed tax schedule. The unobserved choice probabilities P[c(Dm ) = (Y, L)*Jmx ], d 0Dm , (m, x) 0 N are
related to ðN through the linear equations
22
3 1[ck(Dm ) = (Y, L)]@ðNk,
(13) P[c(Dm ) = (Y, L)*Jmx ] =
(Y, L) 0 Dm , (m, x) 0 N.
k0K
The feasible preference distributions are the elements of Ç(ðN ). Hence, the identification region for the
unobserved choice probabilities is
(14) Ç{P[c(Dm ) = (Y, L)*Jmx ], (Y, L) 0 Dm , (m, x) 0 N}
= { { 3 1[ck(Dm ) = (Y, L)]@pk, (Y, L) 0 Dm , (m, x) 0 N}, all (pk, k 0 K) 0 Ç(ðN )}} .
k0K
A question of policy interest is to predict tax revenue under the proposed tax schedule. Recall that
LjT denotes the leisure that person j would choose under the proposed schedule RjT(@). Let the random
variable RT(LT) denote the tax paid by a randomly drawn member of N. Then the per capita tax paid by the
persons in N under the proposed schedule is E[RT (LT )*JN ]. This is related to the preference distribution as
follows:
(15) E[RT (LT )*JN ] =
3 P(m, x)
3 RmT (L) 3 1[ck(Dm ) = (Y, L)]@ðNk,
(m, x) 0 N
(Y, L) 0 D m
k0 K
where RmT(@) is the common tax schedule faced by members of tax group m and where P(m, x) is the fraction
of the population who belong to group (m, x). Hence, the identification region for per capita tax revenue in
class N is the set
(16) Ç{E[RT(LT)*JN ]} = { 3 P(m, x)
(m, x) 0 N
3
RmT(L)
(Y, L) 0 D m
3 1[ck(Dm ) = (Y, L)]@pk,
k0 K
all (pk, k 0 K) 0 Ç(ðN )}.
23
When ÐN places linear restrictions on ðN , convexity of Ç(ðN ) and the linearity of equation (14) in ðN
imply that Ç{E[RT(LT)]}is an interval. The lower and upper bounds of this interval solve two linear
programming problems. The lower bound solves
(17)
min
(p k, k 0 K) 0 Ç(ð N)
3 P(m, x)
(m, x) 0 N
3
RmT(L)
(Y, L) 0 D m
3 1[ck(Dm ) = (Y, L)]@pk.
k0 K
The upper bound solves the analogous problem, with max replacing min.
When the restrictions imposed by ÐN are not linear, minimization problem (16) still gives the sharp
lower bound and the corresponding maximization problem gives the sharp upper bound. However, the
identification region may or may not be the entire interval connecting these bounds, depending on the
structure of ÐN .
3.3. Discussion
The identification analysis of Section 3.2 is mathematically simple. When ÐN places linear
restrictions on ðN , computation is tractable as well. However, two matters are less straightforward, one
analytical and the other substantive.
Analytically, one would like to be able to characterize succinctly the identifying power of alternative
assumptions. When ÐN places linear restrictions on ðN , one can crudely count the number of linear equations
and inequalities that (10), (11), and (12) place on the *K*-dimensional vector ðN of multinomial probabilities.
Unfortunately, doing so does not reveal much about the size and shape of Ç(ðN ), nor about the width of
bounds on derived quantities such as tax revenue under proposed tax schedules. The structure of Ç(ðN )
depends delicately on the choice sets that the various tax groups in class N face under the status quo tax
policy. The implied bounds on outcomes under a proposed tax schedule depend as well on the choice sets
24
that arise under the proposed schedule.
To learn the identifying power of alternative assumptions, it appears necessary to impose them and
compute the resulting identification regions. There does not seem to be an effective shortcut to characterize
Ç(ðN ) or derived quantities. This should not be surprising. A body of research that began with Marschak
(1960) present has found that even simple configurations of observed binary choice probabilities imply subtle
restrictions on preference distributions. See Fishburn (1992) and McFadden (2005).
The substantive issue is that we lack a credible basis for placing informative restrictions on
preference distributions. Standard economic theory only suggests that persons have preference orderings
that satisfy more-is-better. This is a compelling starting point for analysis, but it is not clear how to go
beyond it. The strong assumptions regarding the homogeneity and structure of preferences used in empirical
research on labor supply have been motivated by a desire to obtain tractable point estimates, not by empirical
evidence.
Researchers sometimes try to motivate preference assumptions by viewing time allocation as a
neoclassical problem of budget allocation, with income and leisure being goods akin to apples and oranges.
They may thus introspect on whether leisure is a normal good, whether the income effect of a change in net
wage outweighs the substitution effect, and so on. Such introspection may perhaps be a helpful way to
conjecture plausible forms for income-leisure preferences. However, it is important to understand that the
analogy of time allocation and neoclassical budget allocation is inexact.
The neoclassical problem presumes that a person allocates a specified money budget between two
goods with linear pricing determining the set of feasible allocations. In contrast, the time-allocation problem
presumes that a person begins with two asymmetric endowments, one of time and the other of unearned
income. The asymmetry is that the person can transform time into income through paid work, but he cannot
transform unearned income into time. Moreover, non-linear pricing determines the set of feasible
25
transformations of time into income except in the special case of a proportional tax.12 Further distinctions
arise if the set of feasible time allocations is finite rather than a continuum as assumed in neoclassical theory.
All of these factors make me wary of using neoclassical consumer theory to introspect about time allocation.
3.4. A Computational Experiment
The classical static model is too simplistic to warrant application in substantive study of labor
supply. Nevertheless, it is revealing to perform computational experiments that show the identifying power
of alternative assumptions on the preference distribution. I describe a representative such experiment here.
Given data on labor supply under a status-quo progressive tax schedule, the problem is to predict the
tax revenue per capita that would materialize under a proposed proportional schedule. I show the predictive
power of a sequence of increasingly strong assumptions: (1) more is better; (2) additionally, persons in
specified wage groups have the same distribution of preferences; (3) additionally, preferences have the CES
form; and (4) additionally, all CES utility functions in the specified wage groups have the same elasticity of
substitution.
Tax Policies, Wages, Choice Sets, and Preferences
The progressive status quo policy taxes income at rate 20 percent up to $100,000 per year and at rate
30 percent above $100,000. The proposed proportional policy taxes all income at rate 25 percent. Thus, the
tax schedules cross when gross income equals $200,000, when both take $50,000 tax and yield net income
12
Beginning with Hall (1973), it has been common to model non-proportional tax schedules as if
they are proportional. The idea is to view an individual facing a non-proportional status quo schedule as if
he were facing a proportional tax equal to the marginal tax rate at his chosen time allocation. His
inframarginal net income is treated as a “virtual” form of unearned income. This modeling strategy has
become so prevalent that researchers often neglect to cite its maintained assumptions. These include that
income-leisure preferences have convex indifference curves and that all time allocations in the [0, 1] interval
be feasible.
26
$150,000.
The population is composed of persons who have no unearned income and who have one of eight
annual wages for full-time work: {50, 100, 150, 200, 250, 300, 350, 400} thousand dollars per year. The
assumed population wage distribution is
P(w = 50) = 0.70,
P(w = 100) = 0.20,
P(w = 150) = 0.05,
P(w = 200) = 0.02
P(w = 250) = 0.01,
P(w = 300) = 0.0075, P(w = 350) = 0.0075, P(w = 400) = 0.005.
This distribution roughly approximates that of the 2011 population of adult males in the United States. The
tax schedule under the status quo policy is a simplified idealization of the prevailing American tax schedule.
I suppose that each person can work full-time, three-quarter time, half-time, or not at all. Thus, Ëj
= {0, ¼, ½, 1} for all j 0 J. Appendix Table A1 gives the resulting universe of potential values of (Y, L).
When using the assumption that persons have CES preferences, I suppose that person j has a utility
function of the form
(18)
Uj(Y, L) = [áj(Y/400,000)ñ + (1 ! áj)Lñ ]1/ñ .
j
j
j
Dividing Y by 400,000, which is the maximum annual wage for full-time work, re-scales income to the
interval [0, 1], to make income and leisure have comparable range. The identification analysis presumes that
the parameter á may take one of the 101 values {0, 0.01, 0.02, . . . , 0.99, 1} and that ñ may take one of the
211 values {!100, !90, . . . , !20, !10, !1, !0.99, !0.98, . . . , 0.98, 0.99, 1}. Hence, there are 101 × 211
= 21,311 possible CES preferences. Income and leisure are essentially perfect complements when ñ = !100.
They are perfect substitutes when ñ = 1. The value ñ = 0 gives Cobb-Douglas preferences.
The final step in specification of the experiment is to choose the actual distribution of preferences
27
in the population. I suppose that the population actually contains persons having twenty distinct CES
preferences, with (á, ñ) 0 {0.25, 0.5, 0. 65, 0.75} × {!100, !0.5, 0, 0.5, 1}. Moreover, I suppose that the
population distribution of (á, ñ) is uniform conditional on wage. Thus, the population subgroups with
different wages share the same preference distribution, placing probability 1/20 on each feasible (á, ñ) pair.
This distribution of (á, ñ) values was selected so the population would contain a modest fraction of persons
who choose different time allocations under policies S and T.13
Findings
Given this specification, actual per capita tax revenues under the status quo and proposed tax policies
are $7,652 and $9,211 respectively. Appendix Table A2 details the labor supply and taxes paid under the
two policies by persons with different wages.
If one were to know the wage distribution but not know status quo labor supply, one would only be
able to predict that tax revenue per capita under the proposed policy must lie in the interval [0, $18969]. The
upper and lower bounds are the taxes paid if everyone were to work full time or not at all, respectively.
Given data on status quo labor supply, one’s ability to predict tax revenue under the proposed policy
depends on the assumptions imposed on preferences. The basic assumption that more-is-better yields the
revenue bound [$593, $18969]. This result follows from the analysis of Section 2.3. The upper bound is the
same as the upper bound with no labor supply data because working full time under the proposed policy is
always consistent with the assumption that more-is-better. The lower bound is larger than zero because any
person j for whom LjS # L*j would not choose LjT > L*j . In the context of the computational experiment, this
condition only raises the lower bound on per capita tax revenue from 0 to $593.
I next combine the assumption that more-is-better with the assumption that persons within each of
Persons with the six (á, ñ) pairs {(0.5, !0.5), (0.65, !0.5), (0.65, 0), (0.65, 0.5), (0.65, 1), (0.75,
0.5)} change their choices if they have certain values of w. Those with the other fourteen (á, ñ) pairs never
change regardless of w.
13
28
two broad wage groups have the same distribution of preferences. I assume that persons with w # $200,000
have the same preference distribution and that those with w > $200,000 have the same preference
distribution. I do not assume that persons with wage below and above $200,000 have the same preference
distribution. The resulting bound on per capita tax revenue is [$3744, $14149]. This result follows from
application of equation (16) to each of the two wage groups and aggregation of the findings across the two
groups.14
The above result applies a version of Assumption 1 but does not use any version of Assumption 2.
I next suppose that all persons have CES utility functions. Then application of equation (16) to each wage
group and aggregation across the two groups yields the bound [$6883, $10444] on per capita tax revenue.15
The assumption that everyone has CES preferences is strong. However, it is still much weaker than
the assumptions of homogeneous preferences that have been traditionally maintained in parametric empirical
analyses of labor supply. My final step is to assume that the persons in each broad wage group have the same
elasticity of substitution. Thus, I assume that all persons with w # 200000 have the same value of ñ in their
CES utility functions and, similarly, all persons with w > 200000 have the same ñ. I place no homogeneity
assumption on personal values of á. It turns out that assuming homogeneity of ñ yields an empty
identification region for the preference distribution within the second broad wage group. Thus, the
assumption of homogeneous ñ within each wage group is rejected.
Within the groups with w # $200,000 and w > $200,000, the bounds on per capita tax revenue are
[$2941, $12455] and [$29701, $68906] respectively. The fractions of the population with wages in these
two groups are 0.97 and 0.03 respectively. Averaging the bounds with these fractional weights yields the
aggregate result [$3744, $14149].
14
Within the groups with w # $200,000 and w > $200,000, the bounds on per capita tax revenue are
[$5947, $9143] and [$37148, $52500] respectively. The fractions of the population with wages in these two
groups are 0.97 and 0.03 respectively. Averaging the bounds with these fractional weights yields the
aggregate result [$6883, $10444].
15
29
Computation of the Bounds
Computation of the above bounds ranges from easy to challenging. The bound making only the basic
assumption that more-is-better is simple to compute. It uses only the analytical result of Section 2.3 and does
not require the identification analysis of Section 3.2.
The bound assuming that preferences have the CES form uses the analysis of Section 3.2, but it is
not hard to compute because the number of distinct choice functions is fairly small. Although persons may
have 21,311 distinct values of (á, ñ), this set partitions into a much smaller number of equivalency classes,
within which distinct (á, ñ) values yield the same choices under both tax schedules. It turns out that 108 and
99 equivalency classes suffice to describe the behavior of persons with w # $200,000 and w > $200,000
respectively. Thus, the relevant multinomial distributions ð place probability on 108 and 99 mass points
respectively. Solution of the resulting linear programming problem (17) is tractable.
The most challenging task is to compute the intermediate bound assuming that persons within each
wage group have the same distribution of preferences, but not assuming that preferences have the CES form.
Inspection of Table A1 shows that the members of each wage group face 24 distinct (Y, L) pairs under the
status quo and proposed tax schedules. Thus, the total number of strict preference orderings in each group
is 24! – 6.2 × 1023 . Laborious application of an algorithm that determines and excludes orderings
inconsistent with the more-is-better assumption drastically reduces the number of feasible orderings to 7, 654
and 4,000 in the groups with w # $200,000 and w > $200,000 respectively. Hence, the relevant multinomial
distributions ð place probability on 7,654 and 4,000 mass points respectively. Solution of the resulting linear
programming problem (17) is still tractable.16
16
I am grateful to Jörg Stoye for making available the MATLAB code used to compute the
intermediate bounds. This code was developed for his analysis of related testing and prediction problems
that arise in applications of neoclassical consumer theory. See Kitamura and Stoye (2011).
30
4. Implications for Policy Evaluation
A familiar exercise in normative public economics poses a utilitarian social welfare function and
ranks tax policies by the welfare they achieve. Performing this ranking requires knowledge of income-leisure
preferences both to predict tax revenues and to compute the welfare achieved by alternative policies. Thus,
Mirrlees (1971) wrote in his introductory section (p. 176): “The State is supposed to have perfect information
about the individuals in the economy, their utilities and, consequently, their actions.” This assumption has
been standard in the subsequent literature on optimal income taxation.
Consider a status-quo policy S with tax schedules [RjS(@), j 0 J] and a proposed policy T with
corresponding [RjT(@), j 0 J]. Given cardinal representations for persons’ ordinal utility functions, these
policies yield utilitarian welfare E{[U[(YS(LS), LS]} and E{U[YT(LT), LT]} respectively. Policy T is strictly
preferred to S if
(19) E{[U[(YT (LT ), LT ]} ! E{U[YS(LS), LS]} > 0.
Based on the analysis in Sections 2 and 3, I conclude that we lack the knowledge of preferences
necessary to credibly rank policies. Considering the classical static model, Section 2 showed that basic
revealed preference analysis has little power to predict labor supply under proposed policies. Importantly,
it does not predict whether increasing tax rates reduces or increases work effort. Section 3 showed that
Assumptions 1 and 2 add predictive power, but credible versions of these assumptions generally yield bounds
rather than point predictions. The computational experiment of Section 3.4 is only an illustration, but I think
it typifies the situations that empirical researchers using the classical static model face in practice. Studies
using more general models of time allocation face more difficult inferential problems.
Partial identification of income-leisure preferences creates two distinct difficulties for policy
31
evaluation. First, partial knowledge of the tax revenue yielded by a proposed policy makes it impossible to
ensure that the policy yields a balanced budget. Whatever the intent ex ante, a budget surplus or deficit may
materialize ex post. This makes it infeasible to evaluate policies within a deterministic static framework of
the type used by Mirrlees and others. One requires a suitable dynamic perspective on private time-allocation
and social planning under uncertainty.
Second, partial identification of preferences implies partial knowledge of the utilitarian welfare
achieved by a policy. Thus, choice of tax policy becomes a problem of planning under ambiguity. Basic
decision theory prescribes that a planner should first eliminate dominated policies and then use some
reasonable criterion to choose among the undominated policies. Studying other planning problems, I have
found it instructive to compare the policy choices that result from maximization of subjective expected
welfare and from application of the maximin and minimax-regret criteria (Manski, 2011). A utilitarian
planner may use these or other criteria to choose a tax policy.
A challenge for future research is to develop tractable approaches to the study of choice of tax policy
under ambiguity. One issue, already mentioned above, is the need for a dynamic perspective on planning.
Another is that the knowledge of preferences required to compute welfare is more extensive than the
knowledge needed to predict labor supply. The transformation from preferences to time-allocation under
status quo and proposed tax policies is many-to-one. That is, distinct preferences may imply the same
choices and, hence, be equivalent from the perspective of prediction of labor supply.17 However, such
distinct preferences may have different utilitarian welfare implications.
Yet another complication is that specification of a welfare function requires that the planner select
cardinal representations of ordinal individual utility functions.
These representations are already
consequential for policy comparison in the Mirrlees setting where all members of the population have the
17
The computational experiment in Section 3.4 gave an example. Recall that the 21,311 distinct
values of the CES preference parameters (á, ñ) partitioned into a much smaller number of 108 or 99
equivalency classes, within which distinct (á, ñ) values yielded the same choices under both tax schedules.
32
same known preferences. They are even more consequential when preferences may be heterogeneous and
the planner has only partial knowledge of the preference distribution.
Enriching the Data for Identification of Preferences
I have in this paper assumed that the available data are observations of time allocation under a status
quo tax policy. Collection of richer data can potentially sharpen knowledge of income-leisure preferences.
One avenue, although a difficult one to operationalize, is to resume the performance of randomized
experiments with tax policy that began with the negative income tax experiments of the 1970s. See Hausman
and Wise (1985) and Munnell (1986) for assessment of the lessons learned. Randomized experiments can
contribute to inference by making it credible to assume that groups of persons who face different tax
schedules have the same distribution of income-leisure preferences. Thus, experiments can legitimize
applications of Assumption 1.18
A second avenue is to obtain richer data from individuals than observation of their time allocation
under a single status quo tax policy. If one takes seriously the classical static model, it is useful to repeatedly
observe the labor supply of an individual in circumstances with varying wages, unearned income, or tax
schedules. Assuming that the person’s preferences do not vary over time, transitivity implies that basic
revealed preference analysis of the type performed in Section 2 has increasing predictive power as more
choices are observed (Samuelson, 1938, 1948; Afriat, 1967). However, this interpretation of repeated
observations rests critically on the static model. Suppose, to the contrary, that one views labor supply in
18
Randomized experiments may also be used to directly learn labor supply under proposed policies,
without reference to economic theory. This is the traditional use of experiments in analysis of treatment
response. However, such analysis of experimental data yields no ability to extrapolate findings to policies
not administered in the experiment. Moreover, it provides none of the knowledge of preferences necessary
to compute utilitarian welfare.
33
dynamic terms, as in MaCurdy (1985). Then repeated observations of labor supply provide data on a single
sequential choice path rather than data on choice from multiple independent choice sets.
An alternative way to obtain richer data from individuals is to interview them, pose scenarios with
hypothetical wages or tax schedules, and ask them to predict their time allocation in these scenarios.
Analysis of stated-preference data of this sort has a long history in empirical choice analysis, illustrative
studies using various methods ranging from Beggs, Cardell, and Hausman (1981) and Ben Akiva and
Morikawa (1990) to Blass, Lach, and Manski (2010). The presumption in these and other studies using
stated preference data is that individuals introspect sufficiently well as to provide informative reports of their
preferences. As far as I am aware, the approach has not been used to study income-leisure preferences.
34
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37
Table A1: Universe of (Y, L) Values in the Computational Experiment
Wage
(Net Income, Leisure)
Status Quo
Proposed
Wage
(Net Income, Leisure)
Status Quo
Proposed
50000
[40000, 0]
[30000 ¼]
[20000 ½]
[0, 1]
[37500, 0]
[28125, ¼]
[18750, ½]
[0, 1]
250000
[185000, 0]
[141250, ¼]
[97500, ½]
[0, 1]
[187500, 0]
[140625, ¼]
[93750, ½]
[0, 1]
100000
[80000, 0]
[60000, ¼]
[40000, ½]
[0, 1]
[75000, 0]
[56250, ¼]
[37500, ½]
[0, 1]
300000
[220000 0]
[167500, ¼]
[115000, ½]
[0, 1]
[225000, 0]
[168750, ¼]
[112500, ½]
[0, 1]
150000
[115000, 0]
[88750, ¼]
[60000, ½]
[0, 1]
[112500, 0]
[84375, ¼]
[56250, ½]
[0, 1]
350000
[255000, 0]
[193750, ¼]
[132500, ½]
[0, 1]
[262500, 0]
[196875, ¼]
[131250, ½]
[0, 1]
200000
[150000, 0]
[115000, ¼]
[80000, ½]
[0, 1]
[150000, 0]
[112500, ¼]
[75000, ½]
[0, 1]
400000
[290000, 0]
[220000, ¼]
[150000, ½]
[0, 1]
[300000, 0]
[225000, ¼]
[150000, ½]
[0, 1]
38
Table A2: Actual Labor Supply and Tax Revenues in the Computational Experiment
Wage
Status Quo Policy
Leisure Choice
Proposed Policy
Tax per
Capita
Leisure Choice
L=0
L=¼ L=½
L=1
Tax per
Capita
L=0
L=¼
L=½
L=1
50,000
0
0.45
0.25
0.30
4,625
0
0.45
0.25
0.30
5,781
100,000
0
0.45
0.30
0.25
9,750
0
0.45
0.30
0.25
12,188
150,000
0
0.45
0.30
0.25
15,188
0
0.45
0.30
0.25
18,281
200,000
0.05
0.40
0.35
0.20
23,500
0.05
0.45
0.30
0.20
26,875
250,000
0.05
0.40
0.35
0.20
31,375
0.05
0.45
0.30
0.20
33,594
300,000
0.05
0.25
0.55
0.15
37,625
0.10
0.30
0.45
0.15
41,250
350,000
0.10
0.25
0.50
0.15
47,938
0.10
0.30
0.45
0.15
48,125
400,000
0.10
0.25
0.50
0.15
56,000
0.10
0.25
0.50
0.15
53,750